#7720388

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#37658	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5105 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

(1,2,5,3)(4,7,8,6)(9,10)(11,16)(12,14)(13,15)(18,23)(21,25)(24,30)(28,31);(9,11,13,12)(10,14,15,16)(17,18,22,23)(19,24,29,30)(20,21,26,25)(27,28,32,31);(2,3)(6,7)(10,15)(14,16)(18,23)(20,26)(24,30)(27,32);(1,3,5,2)(4,7,8,6)(9,10)(11,14)(12,16)(13,15)(17,19,22,29)(18,24,23,30)(20,27,26,32)(21,28,25,31);(1,4)(2,6)(3,7)(5,8)(9,12,13,11)(10,16,15,14)(17,20,22,26)(18,25,23,21)(19,27,29,32)(24,31,30,28);(17,21,22,25)(18,26,23,20)(19,28,29,31)(24,32,30,27)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,d^2=gh,e^2=g,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7627584);